array of equations
In this state the only remaining processes are first: to transfer the value which is onV_13,to{V}_{24};and secondly to reduceV_6,V_7,V_13to zero, and to add[30]onetoV_3,in order that the engine may be ready to commence computingB_9.Operations 24 and 25 accomplish these purposes. It may be thought anomalous that Operation 25 is represented as leaving the upper index ofV_3still = unity. But it must be remembered that these indices always begin anew for a separate calculation, and that Operation 25 places uponV_3,thefirstvaluefor the new calculation.
It should be remarked, that when the group (13 ... 23) isrepeated, changes occur in some of theupperindices during the course of the repetition: for example,^3V_6,would become^4V_6,and^5V_6.
We thus see that whenn=1,nine Operation-cards are used; that whenn=2,fourteen Operation-cards are used; and that whenn>2,twenty-five Operation-cards are used; but that no more are needed, however greatnmay be; and not only this, but that these same twenty-five cards suffice for the successive computation of all the Numbers fromB_1,to{B}_{2n - 1},inclusive. With respect to the number ofVariable-cards, it will be remembered, from the explanations in previous Notes, that an average of three such cards to eachoperation(not however to each Operation-card) is the estimate. According to this the computation ofB_1will require twenty-seven Variable-cards;B_3forty-two such cards;B_5seventy-five; and for every succeedingBafterB_5,there would be thirty-three additional Variable-cards (since each repetition of the group (13 ... 23) adds eleven to the number of operations required for computing the previousB). But we must now explain, that whenever there is acycle of operations, and if these merely require to be supplied with numbers from thesame pairs of columnsand likewise each operation to place itsresulton thesamecolumn for every repetition of the whole group, the process then admits of acycle of Variable-cardsfor effecting its purposes. There is obviously much more symmetry and simplicity in the arrangements, when cases do admit of repeating the Variable as well as the Operation-cards. Our present example is of this nature. The only exception to aperfect identityinallthe processes and columns used, for every repetition of Operations (13 ... 23) is, that Operation 21 always requires one of its factors from a new column, and Operation 24 always puts its result on a new column. But as thesevariations follow the same law at each repetition, (Operation 21 always requiring its factor from a column one in advance of that which it used the previous time, and Operation 24 always putting its result on the columnonein advance of that which received the previous result), they are easily provided for in arranging the recurring group (or cycle) of Variable-cards.
We may here remark that the average estimate of three Variable-cards coming into use to each operation, is not to be taken as an absolutely and literally correct amount for all cases and circumstances. Many special circumstances, either in the nature of a problem, or in the arrangements of the engine under certain contingencies, influence and modify this average to a greater or less extent. But it is a very safe and correctgeneralrule to go upon. In the preceding case it will give us seventy-five Variable-cards as the total number which will be necessary for computing anyBafterB_3.This is very nearly the precise amount really used, but we cannot here enter into the minutiæ of the few particular circumstances which occur in this example (as indeed at some one stage or other of probably most computations) to modify slightly this number.
It will be obvious that the verysameseventy-five Variable-cards may be repeated for the computation of every succeeding Number, just on the same principle as admits of the repetition of the thirty-three Variable-cards of Operations (13 ... 23) in the computation of anyoneNumber. Thus there will be acycle of a cycleof Variable-cards.
If we now apply the notation for cycles, as explained inNote E, we may express the operations for computing the Numbers of Bernoulli in the following manner:—array of equationsAgain,array of equationrepresents the total operations for computing every number in succession, fromB_1to{B}_{2n-1}inclusive.
In this formula we see avarying cycleof thefirstorder, and an ordinary cycle of thesecondorder. The latter cycle in this case includes in it the varying cycle.
On inspecting the ten Working-Variables of the diagram, it will be perceived, that although thevalueon any one of them (exceptingV_4,andV_5)goes through a series of changes, theofficewhich each performs is in this calculationfixedandinvariable. ThusV_6always prepares thenumeratorsof the factors of anyA;V_7thedenominators.V_8always receives the (2n-3)th factor of{A}_{2n-1},andV_9the (2n-1)th.V_10always decides which of two courses the succeeding processes are to follow, by feeling for the value ofnthrough means of a subtraction; and so on; but we shall not enumerate further. It is desirable in all calculations, so to arrange the processes, that theofficesperformed by the Variables may be as uniform and fixed as possible.
Diagram for the computation by the Engine of the Numbers of Bernoulli.SeeNote G.(page 67et seq.)
Supposing that it was desired not only to tabulateB_1,B_1,&c., butA_0,A_1,&c.; we have only then to appoint another series of Variables,V_41,V_42,&c., for receiving these latter results as they are successively produced uponV_11.Or again, we may, instead of this, or in addition to this second series of results, wish to tabulate the value of each successivetotalterm of the series (8), viz:A_0,{A}_1{B}_1,{A}_3{B}_3,&c. We have then merely to multiply eachBwith each correspondingA,as produced; and to place these successive products on Result-columns appointed for the purpose.
The formula (8.) is interesting in another point of view. It is one particular case of the general Integral of the following Equation of Mixed Differences:—{d^{2}}/{d x^{2}}(z_{n+1} x^{2 n+2})=(2n+1)(2n+2) z^{n} x^{2n}for certain special suppositions respectingz,xandn.
Thegeneralintegral itself is of the form,z_n=f(n).x+f_1(n)+f_2(n).x^{-1}+f_3(n).x^{-3}+...and it is worthy of remark, that the engine might (in a manner more or less similar to the preceding) calculate the value of this formula upon mostotherhypotheses for the functions in the integral, with as much, or (in many cases) with more, ease than it can formula (8.).
A. A. L.